Teoria dos feixes




Em topologia, um pré-feixe F{displaystyle {mathcal {F}}}mathcal{F} em um espaço topológico X é um funtor contravariante da categoria dos abertos de X numa categoria C{displaystyle {mathcal {C}}}mathcal{C}.


A categoria dos abertos de um espaço topológico tem como morfismos as inclusões, e estes são levados por um feixe nos chamados morfismos de restrição. Caso o pré-feixe satisfaça duas propriedades de colagem, então ele é chamado de feixe. Geralmente a categoria C{displaystyle {mathcal {C}}}mathcal{C} é a categoria Ab, cujos objetos são grupos abelianos e cujos morfismos são homorfismos.


Dizemos que os elementos de F(U){displaystyle {mathcal {F}}(U)}{displaystyle {mathcal {F}}(U)} são as seções do abertos U{displaystyle U}U. Sejam U{displaystyle U}U e V{displaystyle V}V dois abertos de X{displaystyle X}X, e iU,V{displaystyle i_{U,V}}{displaystyle i_{U,V}} o morfismo inclusão de U{displaystyle U}U em V{displaystyle V}V. A imagem de iU,V{displaystyle i_{U,V}}{displaystyle i_{U,V}} por F{displaystyle {mathcal {F}}}mathcal{F} aplicada num elemento s{displaystyle s}s de V{displaystyle {mathcal {V}}}{displaystyle {mathcal {V}}} é chamada de restrição de s a U.





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